Let $X_t, t \in T$ denote a centered Gaussian process. Let $d(t, s) = \sqrt{\mathbb{E} (X_t - X_s)^2}$. Consider a mean function $t \mapsto \mu_t$.

Define the expected supremum $$ S(T, \mu) = \mathbb{E} \Big[\sup_{t \in T} (X_t + \mu_t)\Big]. $$ When $\mu \equiv 0$, Talagrand and others have provided sharp estimates of $S(T, 0)$ in the sense that $$ \frac{1}{C}\, \gamma(T, d) \leq S(T, 0) \leq C \, \gamma(T, d). $$ Above, $C > 0$ is a constant which is universal in the sense that it is independent of $(T, d)$. The exact form of the functional $\gamma$ can be found in the reference [1].

When $\mu_t \neq 0$, we evidently have $$ S(T, \mu) \leq C\, \gamma (T, d) + \sup_{t \in T} \mu_t. $$ However, it is possible to construct examples such that this inequality cannot be reversed.

What results are available in the uncentered case, when $\mu_t \neq 0$?

[1]: Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems, M. Talagrand, 2021.